Bihermitian Structures on Complex Surfaces

Bihermitian complex surfaces are oriented conformal four‐manifolds admitting two independent compatible complex structures. Non‐anti‐self‐dual bihermitian structures on R 4 and the four‐dimensional torus T 4 have recently been discovered by P. Kobak. We show that an oriented compact 4‐manifold, admitting a non‐anti‐self‐dual bihermitian structure, is a torus or K3 surface in the strongly bihermitian case (when the two complex structures are independent at each point) or, otherwise, must be obtained from the complex projective plane or a minimal ruled surface of genus less than 2 by blowing up points along some anti‐canonical divisor (but the actual existence of bihermitian structures in the latter case is still an open question). The paper includes a general method for constructing non‐anti‐self‐dual bihermitian structures on tori, K3 surfaces and S 1 × S 3 . Further properties of compact bihermitian surfaces are also investigated. 1991 Mathematics Subject Classification : 53C12, 53C55, 32J15.